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Full text of "Treatise On Analysis Vol-Ii"

248       XIV    INTEGRATION IN  LOCALLY COMPACT GROUPS

and if t e S and .s e V0, then ,yf e V0 S, which is compact. The restriction
/| V0 S is uniformly continuous and therefore we may take W so that the
relation seVfs0 implies I/CO -/Oo 01 ^e/N(S) for all teS; as before,
we take V = V0 n Vis0 .

Now let g be any function belonging to Jfc(G)- Then the function
(s, i)\-*f(s)g(ts) is continuous on G x G and has compact support. By
(13.21.7) and the left-invariance of \JL, we have

)( [

and since n(f) ^ 0 by hypothesis, it follows that

This shows first of all that Df does not depend on/, for if/' is another function
in jf C(G) such that /t(/') ^ 0, then it follows from above that Dr  ^ =
Dj,  ju, and hence (13.15.3) that Dr and Df> are equal almost everywhere with
respect to /i, But because ju 7^ 0 and is invariant, its support is the whole of G;
also we have seen that Df and Df, are continuous on G; hence the set of
s e G at which Df(s) ^ Dj^s) is open and negligible, therefore empty. In
other words, Df = Dr = D, say. We have therefore, by the definition of the
function D,

for every /e Jf C(G) such that /x(/) ^ 0. This formula, being true in the
complement of a hyperplane in Jfc(G), is true in the whole of .#" C(G)> since
both sides are linear forms in /. Since v ^ 0 we have D(e) ^ 0; therefore /x
and v are proportional, and the proof of (14.1.5) is complete.            Q.E.D.

Any left (resp. right) invariant positive nonzero measure on G is called a
left (resp. right) Haar measure on G. From (14.1.5), any two left (resp. right)
Haar measures on G are proportional.0, we have to find a neighborhood